Proving this seems difficult at first, but Cantor’s reasoning was so simple and original that it is worth understanding it. His is a ‘proof by contradiction’. Here we start with a statement that is opposite of what we want to prove and then show that this leads to an inconsistency.

• Let us take the entire set of decimal numbers i.e. all the numbers between [0,1]. This means that the rational and irrational numbers are in this together. And we know that the fractions or the rational numbers are ‘countably infinite’.

The essence of Cantor’s proof is that we assume that the decimals (irrational & rational numbers both put together) are as numerous as the natural numbers ( i.e. integers, and we have shown above that the rational numbers have the same infinity א₀, as the integers themselves) then it should be possible to label them one by one.

Let us accordingly number the list a₁, a₂, a₃, a₄ and so on :-

We imagine each a₁, a₂, a₃   etc. to be a decimal number that continues indefinitely. In case it terminates (i.e. it is rational and is an exact fraction) then we will fill up the digits with zeros.